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What is the connection between the Standard Model gauge group and the critical dimension of string theory?

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(this is a self-answered question, but all are free to answer, obviously)

I've often heard that the critical dimension of 10 in superstring theories is related to the standard model gauge group along with the 4 observed dimensions in which general relativity applies. That is, 3 dimensions from SU(3) Quantum Chromodynamics, 2 from SU(2) the weak interaction, U(1) from electromagnetism, and the four uncompactified dimensions from gravity.

What is the explanation for this similarity, especially the four uncompactified dimensions of gravity, as it isn't very clear what the lack of compactification has to do with this.

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1 Answer

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(EDIT: This answer is completely wrong, SU(2) is diffeomorphic to S3 and SU(3) is obviously not three-dimensional. I came up with this ridiculous explanation based on an incorrectly done analogy to Kaluza-Klein theory, and right now I feel like I wrote it half-braindead. The question remains open, and I'm only leaving this answer on for the sake of the comments.)

It's true that the type of compactification has to do with the observed interactions. The U(1), SU(2), and SU(3) groups appear in the Calabi-Yau or G2 manifolds that the theory is compactified on. The curvature of the manifold with U(1) symmetry is linked to electromagnetism in QED, and analogous arguments apply for the weak and strong interactions. And then there are the uncompactified 4 dimensions that are found in general relativity.

The explanation for this relation is best seen in the supergravity approximation. In Kaluza-Klein theory you have gravity (standard general relativity) in 5-dimensional spacetime, which is compactified around a manifold with U(1) symmetry to obtain 4-dimensional gravity and another force with U(1) gauge symmetry, which we know as electromagnetism.

This is taken further in supergravity, where you have compactification around SU(2) and SU(3) manifolds (requiring 2 and 3 extra dimensions respectively), each giving rise to new forces with SU(2) and SU(3) gauge symmetry respectively.

As supergravity is found in the classical limit of superstring theory, the critical dimension is carried over to string theory.

I don't understand enough about string theory to be sure but your answer sounds somewhat superficial to me. If the standard model with group $U(1) \times SU(2) \times SU(3)$ counts as 6 dimensions and it would appear as a limit of a grand unified $SU(5)$ theory with broken symmetry, wouldn't the critical dimension suddenly go down by one?

I don't think that this answer makes sense. What does it mean that a group "appears in the Calabi-Yau or G_2 manifolds"? What do you call a "SU(2) (or SU(3)) manifold"? To obtain a gauge symmetry of group G by Kaluza-Klein mechanism, one needs to compactify on a manifold with isometry group containing G. There is no compact 3-manifold of isometry group SU(3). Simplest compact manifolds of isometry group SU(3)xSU(2)xU(1) are of dimension 7 (like $\mathbb{CP}^2 \times S^2 \times S^1$) and one could try to compactify 11d supergravity on it (see http://inspirehep.net/record/10244 ) but this has a major drawback if one wants something realistic: it is not possible to obtain chiral gauge coupling by Kaluza-Klein mechanism. Gauge symmetry in string theory does not (in general, and in particular for chiral gauge couplings) come from Kaluza-Klein mechanism but from a gauge group in 10 dimensions in heterotic string or from D-branes constructions in type II string.

@40227 Thanks, you're right. By SU(2) and SU(3) manifolds I was referring to the SU(2) and SU(3) bundles, but I don't know how I messed up with the dimensionality of the bundles, as @Arnold also hinted at. I was unaware of the Witten paper you linked to, too.

Gauge symmetry in string theory does not (in general, and in particular for chiral gauge couplings) come from Kaluza-Klein mechanism but from a gauge group in 10 dimensions in heterotic string or from D-branes constructions in type II string.

I know this applies for string theory, but was under the wrong impression that the Kaluza-Klein explanation applies to supergravity as if being equivalent or dual to the correct explanation.

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